American Journal of Signal Processing 2013, 3(4): 102-105 DOI: 10.5923/j.ajsp.20130304.02 Inverting Non-minimum Phase FIR Transfer Functions T. J. Moir School of Engineering, AUT University, Auckland, New-Zealand AbstractTheapproximatesolutiontotheinverseofanon-minimumphasefinite-impulseresponse(FIR)transfer functionisgiven.Suchaproblemarisesinmanyareaswhichincluderoom-acousticsandchannelequalisation.Unlike previous work the method does not require multiple sensors or the use of all-pass transfer function networks. KeywordsNon-minimum Phase, Inversion, Impulse Response, Signal Processing 1. Introduction FIRtransferfunctionsaremorethanoftentheimpulse responseofacommunicationchanneloracoustic environmentfoundexperimentally.Inthecaseof minimum-phase FIR transfer functions their inverse is easily found,butforthenon-minimumphasecasetheinverse becomesunstable.Previousworkinthearea[1]involves splittingthetransferfunctionintothecombinationofan all-passtransferfunctionsandaminimum-phasetransfer function.Theminimum-phasepartistheninvertedleaving the all-pass transfer function which has magnitude unity (and hence does not effect the magnitude response of the inverse filter).Providedthegroup-delayisconstantthismethodis satisfactory.However,asdiscussedin[1]iftheall-pass transfer function does not have constant group-delay then the conventional method will have a noticeable non-linear phase miss-match. The importance of a linear phase solution is that frequenciesarescaledbyaproportionalamountwhen passingthroughthechannelandfilter,thuspreservingthe shapeoftheoriginalsignal.Linearphaseisequivalentto having a pure time-delay (constant group-delay) which will forexampleintroducenoperceptualdistortioninaspeech signal. Themethodusedhererecognisesthattheinverseofa non-minimumphaseimpulse-responsecaneitherbean unstablesystemorastableyetuncausalsystem.The uncausalpartisthenusedbyemployingatime-delayand shiftingthispartbackintopositivetime.Unfortunatelythe uncausal impulse response has an infinite impulse response soitmusttruncatedmakingthemethodapproximate depending on the length of the delay. This method is not new inprinciple,havingbeenusedfordecadesinadaptive filtering using least-mean squares (LMS) algorithms[2, 3]. In * Corresponding author: Tom.Moir@aut.ac.nz (T. J. Moir) Published online at http://journal.sapub.org/ajsp Copyright 2013 Scientific & Academic Publishing. All Rights Reserved audioacoustics,theequalizationofloudspeaker characteristicsfacessimilarproblems[4]andinsound rendering[5].Inthecontrolliteraturesimilarproblemsare oftenencountered[6].Techniquesalreadyexistfortwoor moresensors[7],butthisexplicitsolutionisnewandonly requires one signal-transmission channel. 2. Mathematical Preliminaries Ifapolynomialdefinedas 1 1 20 1 2( ) ...nnX z x x z x z x z = + + + + ofdegreen with real coefficients has all its roots within the unit circle in the z plane, then it is termed strict sense minimum phase. No zerosareassumedtobeontheunitcircle.Forsimplicity 1( ) X zisoftenwrittenasX ,omittingthecomplex argument 1z.Theconjugatepolynomial * 20 1 2( ) ...nnX z x x z x z x z = + + + + isstrictsense non-minumum phase having all of its roots outside the unit circle on the z plane. The reciprocal polynomial is defined as 1 ( 1) ( 2) *0 1 2( ) ... ( )n n n nnX z x z x z x z x z X z = + + + + =whichhasallitsrootsoutsidetheunitcircleprovided 1( ) X zisstrictsenseminimumphase.ThezerosofX arethezerosofX reflectedintheunitcircle.Similarly * 1 1( ) ( )nX z z X z =has all its roots within the unit circle. For polynomials which are strict sense non-minimum phase, wecanfactorise 1 1 11 2( ) ( ) ( ) X z X z X z = where 1Xisstrictsenseminimumphaseand 2X isstrictsense non-minimum phase. 3. Theory BeginwiththeFIRtransferfunctionofatransmission channel as described by the product of two polynomials American Journal of Signal Processing 2013, 3(4): 102-105103 1 1 11 2( ) ( ) ( ) = H z B z B z (1) where[8]11( )B z isstrictsenseminimumphaseand 12( )B z isstrictsensenon-minimumphase.Their polynomial dimensions are 1nb and 2nb respectively. We require an inverse transfer-function 11 1 11 21 1( )( ) ( ) ( ) = = F zH z B z B z(2) Toproceedfurtherwenotethat * *2 2 2 2= B B B B and substitute *2 22 *2=B BBB into (2) giving *1 2*2 2( )=BF zBB B (3) Now the reciprocal polynomial 2B is stable having all of itsrootswithintheunitcircle,but *2Bisnot.Therefore expand in positive powers of z *2 20 1 2 *2...Bd d z d zB = + + +(4) Foranon-minimumphasepolynomial 2B ofdegree 2nb the expansion (4) is easily found by a simple recursion. For example if 221 22 0 1 2... = + + + +nbnbB z z z (5) then 200 =nbd 22102( ) +==nbi nb i j jjinbdd 21, 2... = i nb (6) and 222101+ == nbi j i j nbjnbd d for 2> i nb (7) Delaying by terms *1 2 20 1 1 2*2... ... ...Bz d z d z d d z d zB ++ += + + + + + + (8) Which is a Laurent series with negative (causal) and positive (uncasual) powers of z. **2*2= +Bz D EB(9) Thereciprocalpolynomial 10 1... += + + + D d z d z d ,whereas * 21 2...+ += + + E d z d z isaninfinite polynomial.However,ifthedelay islargeenoughwecanapproximate *0 E sincethisispartofaconvergent power-series whose higher terms will converge to zero. So (3) becomes 11 2( )DF zB Bfor large (10) which is the inverse filter. It would be possible of course to express this as an FIR transfer function by using polynomial division and further truncation. 4. Example ConsideranFIRtransferfunctionwith1 1 2( ) 1 2.6 1.2 = + H z z z or 1 11( ) 1 0.6 = B z z and1 12( ) 1 2 = B z z . Find its inverse and use a delay of6 = . Solution:Use the fact that*21 2 = B z ,1 *2 22 , 2= + = + B z B z . Now expand *5 6 6 5 4 3 2 1 * 2*21 20.5 0.75 0.375 0.1875 0.0937 0.04687 0.023437 ... E2B zz z z z z z z zB z = = + + + + + + + + So that 6 5 4 3 2 10.5 0.75 0.375 0.1875 0.0937 0.04687 0.0234375 D z z z z z z = + + + + + + Substituting into (10) we find the inverse transfer function as 6 5 4 3 2 111 1( 0.5 0.75 0.375 0.1875 0.0937 0.04687 0.0234375)( ) 0.5(1 0.5 )(1 0.6 )z z z z z zF zz z + + + + + + 104T. J. Moir:Inverting Non-minimum Phase FIR Transfer Functions For comparison purposes we use the alternative approach [1] which employs an all-pass transfer function. 1' 11 1 10.5 (1 2 )( )(1 0.5 )(1 0.6 ) 2(1 0.5 ) = zF zz z z Nowwhentheoriginalchanneltransfer-function 1 1 2( ) 1 2.6 1.2 = + H z z z isappliedbyconvolution withtheinversefilter,unitygainshouldbeachievedupto half-samplingfrequency.Acomparisonofthemagnitude and phase inverse transfer functions are shown in Figs1 and 2 respectively. Fig 1 shows a comparison for different values ofdelayshowingthattheamountofripplereduces considerably as the delay increases. Figure 1.Magnitude and phase of equalised impulse response using new method Figure2.Magnitudeandphaseofequalisedresponseusingall-pass method It can be seen that there is some 0.5dB peak-to peak ripple in the magnitude for the new case when the delay is 4 whilst fortheall-passmethodthemagnitudeisperfectlyflat. However,thephasefortheall-passmethodisnon-linear whereas for the new method the phase is linear. Furthermore the ripple for the new method is further reduced by making the delay term bigger. For example, for a delay of 6 samples thedBrippleisdownto0.1dBpeak-topeakwhilststill maintaining linear phase. 5. Conclusions Asinglesensorapproachtotheinversionof non-minimum phase FIR transfer functions has been shown. The method is computationally simple and does not rely on all-pass transfer functions. Instead, it uses the uncausal part ofthetransferfunctioninverseitselfwhichisshiftedinto positivetimebyusingatime-delay.Thisgivesrisetoan equalisedtransferfunctionwhichhasasmallamountof rippleyetislinearphase.Forsmallimpulseresponsesit becomes quite an easy task to find the inverse and this is of sometheoreticalimportance,However,formuchlarger impulseresponseswhichcouldbemetinanacoustic environment,themethodrequiresthefactorizationoflarge polynomialsintominimumandnon-minimumphase polynomials,whichisinitselfquiteacomputationaltask. Thereforeanefficientcomputationalalgorithmneedstobe found for performing this task without having to find all of the individual roots of the polynomials themselves. REFERENCES [1]S. T. Neely and J. B. Allen, "Invertibility of a room impulse response," J.Acoust.Soc.Amer., vol. 66, pp. 165-169, 1979. [2]S.Haykin,Adaptivefiltertheory.EnglewoodCliffsNew Jersey: Prentice Hall, 1986. [3]B.WidrowandM.Bilello,"AdaptiveInverseControl," presentedattheProceedingsofthe1993International SymposiumonIntelligentControl,Chicago,IllinoisUSA, 1993. [4]A.MarquesandD.Freitas,"Infiniteimpulseresponse(IIR) inversefilterdesignfortheequalizationofnon-minimum phaseloudspeakersystems,"inApplicationsofSignal Processing to Audio and Acoustics, 2005. IEEE Workshop on, 2005, pp. 170-173. [5]A.Mouchtaris,P.Reveliotis,andC.Kyriakakis, "Non-minimumphaseinversefiltermethodsforimmersive audiorendering,"inAcoustics,Speech,andSignal Processing,1999.Proceedings.,1999IEEEInternational Conference on, 1999, pp. 3077-3080 vol.6. [6]B.WidrowandM.Bilello,"Adaptiveinversecontrol,"in IntelligentControl,1993.,Proceedingsofthe1993IEEE International Symposium on, 1993, pp. 1-6. [7]M.MiyoshiandY.Kaneda,"Inversefilteringofroom acoustics,"Acoustics,Speech,andSignalProcessing[see alsoIEEETransactionsonSignalProcessing],IEEE Transactions on, vol. 36, pp. 145-152, 1988. American Journal of Signal Processing 2013, 3(4): 102-105105 [8]T.J.Moir,"Az-domaintransferfunctionsolutiontothe non-minimumphaseacousticbeamformer,"Intern.J.Syst. Sci., vol. 38, pp. 563-575, 2007.